Asymptotic controllability implies feedback stabilization

@article{Clarke1997AsymptoticCI,
  title={Asymptotic controllability implies feedback stabilization},
  author={Frank H. Clarke and Yu. S. Ledyaev and Eduardo Sontag and Andreĭ I. Subbotin},
  journal={IEEE Trans. Autom. Control.},
  year={1997},
  volume={42},
  pages={1394-1407}
}
It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a control-Lyapunov function, iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium. A major technical problem, and one of the contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous… 

Figures from this paper

Asymptotic Controllability and Robust Asymptotic Stabilizability

  • C. Prieur
  • Mathematics
    SIAM J. Control. Optim.
  • 2005
It is proved that the origin of all globally asymptotically controllable systems can be globally asylptotic stabilized via a hybrid feedback with robustness with respect to measurement noise, actuator errors, and external disturbances.

Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks

This work constructs discontinuous feedback laws and makes it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems which are affine in the control.

Sufficient Conditions for Closed-Loop Asymptotic Controllability and Stabilization by Smooth Time-Varying Feedback Integrator

  • J. Tsinias
  • Mathematics
    IEEE Transactions on Automatic Control
  • 2008
The main hypotheses are based on the existence of an almost smooth time-varying control Lyapunov function, and the corresponding results generalize earlier works in the literature concerning dynamic stabilization for autonomous systems.

Asymptotic controllability implies continuous-discrete time feedback stabilizability

In this paper, the relation between asymptotic controllability and feedback stabilizability of general nonlinear systems is investigated. It is proved that asymptotic controllability implies for any

Stability and Feedback Stabilization

  • Eduardo Sontag
  • Mathematics
    Encyclopedia of Complexity and Systems Science
  • 2009
Stability A globally asymptotically stable equilibrium is a state with the property that all solutions converge to this state, with no large excursions. Stabilization A system is stabilizable (with

Converse Lyapunov theorems for control systems with unbounded controls

Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function

  • C. KellettA. Teel
  • Mathematics
    Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
  • 2000
We show that uniform global asymptotic controllability to a closed (not necessarily compact) set for a locally Lipschitz nonlinear control system implies the existence of a locally Lipschitz

Feedback Stabilization and Lyapunov Functions

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, it is established that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.
...

References

SHOWING 1-10 OF 33 REFERENCES

A Lyapunov-Like Characterization of Asymptotic Controllability

It is shown that a control system in ${\bf R}^n $ is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can

Nonsmooth control-Lyapunov functions

It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF

Global asymptotic stabilization for controllable systems without drift

  • J. Coron
  • Mathematics
    Math. Control. Signals Syst.
  • 1992
This paper proves that the accessibility rank condition on ℝn{0} is sufficient to guarantee the existence of a global smooth time-varying (but periodic) feedback stabilizer, for systems without

A REMARK ON ROBUST STABILIZATION OF GENERAL ASYMPTOTICALLY CONTROLLABLE SYSTEMS

It was shown recently by Clarke, Ledyaev, Sontag and Subbotin that any asymptotically controllable system can be stabilized by means of a certain type of discontinuous feedback. The feedback laws

Control of systems without drift via generic loops

A simple numerical technique for the steering of arbitrary analytic systems with no drift is proposed, based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajectories.

Qualitative properties of trajectories of control systems: A survey

We present a unified approach to a complex of related issues in control theory, one based to a great extent on the methods of nonsmooth analysis. The issues include invariance, stability, equilibria,

Asymptotic stability and feedback stabilization

General theorems are established which are strong enough to show that a) there is a continuous control law (u,v) = (u(xry, z) rv(x,y,z)) which makes the origin asympEoticatly stable for x=u y=v z=xy and that b) there exists no continuous control laws.

Remarks on continuous feedback

  • Eduardo SontagH. Sussmann
  • Mathematics
    1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes
  • 1980
We show that, in general, it is impossible to stabilize a controllable system by means of a continuous feedback, even if memory is allowed. No optimality considerations are involved. All state spaces

Stabilization with relaxed controls

On the stabilization in finite time of locally controllable systems by means of continuous time-vary

It is proven that, if, for any positive time $T$, there exists an open-loop control $u(a,t)$ depending continuously on the initial data $a$, vanishing for $a=0$, and steering a small neighborhood of